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1. CJM Online first

Papikian, Mihran; Rabinoff, Joseph
Optimal quotients of Jacobians with toric reduction and component groups
Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J \to E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map $\Phi_J \to \Phi_E$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which $\Phi_J \to \Phi_E$ is surjective, and discuss when these criteria hold for the Jacobians of modular curves.

Keywords:Jacobians with toric reduction, component groups, modular curves
Categories:11G18, 14G22, 14G20

2. CJM Online first

Levinson, Jake
One-dimensional Schubert problems with respect to osculating flags
We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real -- in this case, for zero-dimensional Schubert problems, the solutions are "as real as possible". Recent work by Speyer has extended the theory to the moduli space $ \overline{\mathcal{M}_{0,r}} $, allowing the points to collide. These give rise to smooth covers of $ \overline{\mathcal{M}_{0,r}} (\mathbb{R}) $, with structure and monodromy described by Young tableaux and jeu de taquin. In this paper, we give analogous results on one-dimensional Schubert problems over $ \overline{\mathcal{M}_{0,r}} $. Their (real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over $\mathcal{M}_{0,r}$, our results show that the real points of the solution curves are smooth. We also find a new identity involving "first-order" K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

Keywords:Schubert calculus, stable curves, Shapiro-Shapiro Conjecture, jeu de taquin, growth diagram, promotion
Categories:14N15, 05E99

3. CJM 2016 (vol 68 pp. 1096)

Smith, Benjamin H.
Singular $G$-Monopoles on $S^1\times \Sigma$
This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on $S^1\times \Sigma$ and $\vec{t}$-stable meromorphic pairs on $\Sigma$. A theorem of B. Charbonneau and J. Hurtubise is thus generalized here from unitary to arbitrary compact, connected gauge groups. The required distinctions and similarities for unitary versus arbitrary gauge are clearly outlined and many parallels are drawn for easy transition. Once the correspondence theorem is complete, the spectral decomposition is addressed.

Keywords:connection, curvature, instanton, monopole, stability, Bogomolny equation, Sasakian geometry, cameral covers
Categories:53C07, 14D20

4. CJM 2016 (vol 68 pp. 784)

Doran, Charles F.; Harder, Andrew
Toric Degenerations and Laurent Polynomials Related to Givental's Landau-Ginzburg Models
For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Givental's Landau-Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau-Ginzburg models can be expressed as corresponding Laurent polynomials. We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi-Yau varieties.

Keywords:Fano varieties, Landau-Ginzburg models, Calabi-Yau varieties, toric varieties
Categories:14M25, 14J32, 14J33, 14J45

5. CJM Online first

Moon, Han-Bom
Mori's program for $\overline{M}_{0,7}$ with symmetric divisors
We complete Mori's program with symmetric divisors for the moduli space of stable seven-pointed rational curves. We describe all birational models in terms of explicit blow-ups and blow-downs. We also give a moduli theoretic description of the first flip, which has not appeared in the literature.

Keywords:moduli of curves, minimal model program, Mori dream space
Categories:14H10, 14E30

6. CJM Online first

Garbagnati, Alice
On K3 surface quotients of K3 or Abelian surfaces
The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a K3 surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces which are (rationally) $G$-covered by Abelian or K3 surfaces (in the latter case $G$ is an Abelian group). If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result was already known, so we extend it to the other cases. Moreover, we prove that a K3 surface $X_G$ is the minimal model of the quotient of an Abelian surface by a group $G$ if and only if a certain configuration of rational curves is present on $X_G$. Again this result was known only in some special cases, in particular if $G$ has order 2 or 3.

Keywords:K3 surfaces, Kummer surfaces, Kummer type lattice, quotient surfaces
Categories:14J28, 14J50, 14J10

7. CJM 2016 (vol 68 pp. 541)

Garcia-Armas, Mario
Strongly Incompressible Curves
Let $G$ be a finite group. A faithful $G$-variety $X$ is called strongly incompressible if every dominant $G$-equivariant rational map of $X$ onto another faithful $G$-variety $Y$ is birational. We settle the problem of existence of strongly incompressible $G$-curves for any finite group $G$ and any base field $k$ of characteristic zero.

Keywords:algebraic curves, group actions, Galois cohomology
Categories:14L30, 14E07, 14H37

8. CJM 2016 (vol 68 pp. 504)

Biswas, Indranil; Gómez, Tomás L.; Logares, Marina
Integrable Systems and Torelli Theorems for the Moduli Spaces of Parabolic Bundles and Parabolic Higgs Bundles
We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.

Keywords:parabolic bundle, Higgs field, Torelli theorem
Categories:14D22, 14D20

9. CJM 2016 (vol 68 pp. 361)

Fité, Francesc; González, Josep; Lario, Joan Carles
Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$ of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians $\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell-2$, where $\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by certain subgroups of automorphisms of $\mathcal{C}$. It is well known that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely simple abelian variety $B_k$ with complex multiplication. We call degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM type. For a non-degenerate pair $(\ell,k)$, we compute the Sato-Tate group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of $(\ell,k)$ being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $\ell$-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Keywords:Sato-Tate group, Fermat curve, Frobenius distribution
Categories:11D41, 11M50, 11G10, 14G10

10. CJM 2016 (vol 68 pp. 280)

da Silva, Genival; Kerr, Matt; Pearlstein, Gregory
Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution
We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\leq d\leq6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is $G_{2}$) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains.

Keywords:variation of Hodge structure, limiting mixed Hodge structure, Calabi-Yau variety, middle convolution, Mumford-Tate group
Categories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20

11. CJM 2015 (vol 68 pp. 241)

Allermann, Lars; Hampe, Simon; Rau, Johannes
On Rational Equivalence in Tropical Geometry
This article discusses the concept of rational equivalence in tropical geometry (and replaces an older and imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

Keywords:tropical geometry, rational equivalence
Category:14T05

12. CJM 2015 (vol 68 pp. 67)

Ishida, Hirotaka
A Lower Bound on the Euler-Poincaré Characteristic of Certain Surfaces of General Type with a Linear Pencil of Hyperelliptic Curves
Let $S$ be a surface of general type. In this article, when there exists a relatively minimal hyperelliptic fibration $f \colon S \rightarrow \mathbb{P}^1$ whose slope is less than or equal to four, we show the lower bound on the Euler-Poincaré characteristic of $S$. Furthermore, we prove that our bound is the best possible by giving required hyperelliptic fibrations.

Keywords:hyperelliptic fibration, surface of general type, double cover
Categories:14D05, 14J29, 14H30

13. CJM 2015 (vol 68 pp. 334)

Demchenko, Oleg; Gurevich, Alexander
Kernels in the Category of Formal Group Laws
Fontaine described the category of formal groups over the ring of Witt vectors over a finite field of characteristic $p$ with the aid of triples consisting of the module of logarithms, the Dieudonné module and the morphism from the former to the latter. We propose an explicit construction for the kernels in this category in term of Fontaine's triples. The construction is applied to the formal norm homomorphism in the case of an unramified extension of $\mathbb{Q}_p$ and of a totally ramified extension of degree less or equal than $p$. A similar consideration applied to a global extension allows us to establish the existence of a strict isomorphism between the formal norm torus and a formal group law coming from $L$-series.

Keywords:formal groups, $p$-divisible groups, Dieudonne modules, norm tori
Category:14L05

14. CJM 2015 (vol 67 pp. 961)

Abuaf, Roland; Boralevi, Ada
Orthogonal Bundles and Skew-Hamiltonian Matrices
Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial splitting on the general line, is smooth irreducible of dimension $(r-2)n-\binom{r}{2}$ for $r=n$ and $n \ge 4$, and $r=n-1$ and $n\ge 8$. We speculate that the result holds in greater generality.

Keywords:orthogonal vector bundles, moduli spaces, skew-Hamiltonian matrices
Categories:14J60, 15B99

15. CJM 2015 (vol 67 pp. 1201)

Aluffi, Paolo; Faber, Eleonore
Chern Classes of Splayed Intersections
We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a splayedness assumption. We show that the relation for the Chern-Schwartz-MacPherson classes holds for two splayed hypersurfaces in a nonsingular variety, and under a `strong splayedness' assumption for more general subschemes. Moreover, the relation is shown to hold for the Chern-Fulton classes of any two splayed subschemes. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.

Keywords:splayed intersection, Chern-Schwartz-MacPherson class, Chern-Fulton class, splayed blowup, Segre class
Categories:14C17, 14J17

16. CJM 2015 (vol 67 pp. 1109)

Nohara, Yuichi; Ueda, Kazushi
Goldman Systems and Bending Systems
We show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.

Keywords:toric degeneration
Categories:53D30, 14H60

17. CJM 2015 (vol 68 pp. 24)

Bonfanti, Matteo Alfonso; van Geemen, Bert
Abelian Surfaces with an Automorphism and Quaternionic Multiplication
We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of $(2,4)$-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}[\sqrt{2}]$.

Keywords:abelian surfaces, moduli, shimura curves
Categories:14K10, 11G10, 14K20

18. CJM 2015 (vol 67 pp. 696)

Zhang, Tong
Geography of Irregular Gorenstein 3-folds
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo type inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.

Keywords:3-fold, geography, irregular variety
Category:14J30

19. CJM 2014 (vol 67 pp. 923)

Pan, Ivan Edgardo; Simis, Aron
Cremona Maps of de Jonquières Type
This paper is concerned with suitable generalizations of a plane de Jonquières map to higher dimensional space $\mathbb{P}^n$ with $n\geq 3$. For each given point of $\mathbb{P}^n$ there is a subgroup of the entire Cremona group of dimension $n$ consisting of such maps. One studies both geometric and group-theoretical properties of this notion. In the case where $n=3$ one describes an explicit set of generators of the group and gives a homological characterization of a basic subgroup thereof.

Keywords:Cremona map, de Jonquières map, Cremona group, minimal free resolution
Categories:14E05, 13D02, 13H10, 14E07, 14M05, 14M25

20. CJM 2014 (vol 67 pp. 527)

Brugallé, Erwan; Shaw, Kristin
Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory
We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on a relation between tropical and complex intersection theories which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.

Keywords:tropical geometry, amoebas, approximation of tropical varieties, intersection theory
Categories:14T05, 14M25

21. CJM 2014 (vol 67 pp. 639)

Gonzalez, Jose Luis; Karu, Kalle
Projectivity in Algebraic Cobordism
The algebraic cobordism group of a scheme is generated by cycles that are proper morphisms from smooth quasiprojective varieties. We prove that over a field of characteristic zero the quasiprojectivity assumption can be omitted to get the same theory.

Keywords:algebraic cobordism, quasiprojectivity, cobordism cycles
Categories:14C17, 14F43, 55N22

22. CJM 2014 (vol 67 pp. 893)

Mok, Chung Pang; Tan, Fucheng
Overconvergent Families of Siegel-Hilbert Modular Forms
We construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to construction of Galois representations for automorphic forms of non-cohomological weights.

Keywords:p-adic automorphic form, rigid analytic geometry
Categories:11F46, 14G22

23. CJM 2014 (vol 67 pp. 1219)

Balwe, Chetan
$p$-adic and Motivic Measure on Artin $n$-stacks
We define a notion of $p$-adic measure on Artin $n$-stacks which are of strongly finite type over the ring of $p$-adic integers. $p$-adic measure on schemes can be evaluated by counting points on the reduction of the scheme modulo $p^n$. We show that an analogous construction works in the case of Artin stacks as well if we count the points using the counting measure defined by Toën. As a consequence, we obtain the result that the Poincaré and Serre series of such stacks are rational functions, thus extending Denef's result for varieties. Finally, using motivic integration we show that as $p$ varies, the rationality of the Serre series of an Artin stack defined over the integers is uniform with respect to $p$.

Keywords:p-adic integration, motivic integration, Artin stacks
Categories:14E18, 14A20

24. CJM 2014 (vol 67 pp. 848)

Köck, Bernhard; Tait, Joseph
Faithfulness of Actions on Riemann-Roch Spaces
Given a faithful action of a finite group $G$ on an algebraic curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for the induced action of~$G$ on the Riemann-Roch space~$H^0(X,\mathcal{O}_X(D))$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least~$2g_X-2$. This leads to a concise answer to the question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes m})$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we furthermore provide an explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of~$G$ on the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.

Keywords:faithful action, Riemann-Roch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology
Categories:14H30, 30F30, 14L30, 14D15, 11R32

25. CJM 2014 (vol 67 pp. 55)

Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina
On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ${\mathbb C}$, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ${\mathbb N}$-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of $\mathbb{N}$-graded Lie algebras $L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$ and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the structure of these algebras with the property $L=\langle L_1 \rangle$. In this paper we study those generated by the first and $q$-th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under some technical condition, there can only be one isomorphism type of such algebras. For $q=3$ we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification
Categories:17B70, 14F45
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